<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">V</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="clvect_1.html#V2" title="CLVECT_1:attr.2">ComplexLinearSpace-like</a>  <a href="clvect_1.html#L1" title="CLVECT_1:struct.1">CLSStruct</a> ;<br/><b>let </b><font color="Maroon" title="c2">M1</font>, <font color="Maroon" title="c3">M2</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>;<br/><b>let </b><font color="Maroon" title="c4">z</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>;<br/>





<a NAME="E1:86"/>
for <font color="Olive" title="b1">x</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font></span>)</span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c3">M2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/>

<b>assume </b><a NAME="E1:86_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">w'</font> being   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:86_1"/><i><font color="Green" title="E51">A1</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c6">w'</font> &amp; <font color="Maroon" title="c6">w'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c7">w1</font>, <font color="Maroon" title="c8">w2</font> being   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E5:86_1"/><i><font color="Green" title="E52">A2</font></i>: 
( <font color="Maroon" title="c6">w'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">w1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">w2</font> &amp; <font color="Maroon" title="c7">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M1</font> &amp; <font color="Maroon" title="c8">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M2</font> )
 <b>by </b><i><a class="txt" href="convex4.html#E3:86_1"><i><font color="Green" title="E51">A1</font></i></a></i>;<br/>
<a NAME="E6:86_1"/><i><font color="Green" title="E53">A3</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c7">w1</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c8">w2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="convex4.html#E3:86_1"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="convex4.html#E5:86_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="clvect_1.html#D2" title="CLVECT_1:def.2">CLVECT_1:def 2</a></i>;<br/>
<a NAME="E7:86_1"/>
( <font color="Maroon" title="c4">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c7">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M1</font> &amp; <font color="Maroon" title="c4">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c8">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c3">M2</font> )
 
<b>by </b><i><a class="txt" href="convex4.html#E5:86_1"><i><font color="Green" title="E52">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:86_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c3">M2</font></span>)</span>
 <b>by </b><i><a class="txt" href="convex4.html#E6:86_1"><i><font color="Green" title="E53">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:86"/><b>then </b><i><font color="Green" title="E51">A5</font></i>: 
<font color="Maroon" title="c4">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c3">M2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></i>;<br/>
<a NAME="E3:86"/>
for <font color="Olive" title="b1">x</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c3">M2</font></span>)</span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/>

<b>assume </b><a NAME="E1:86_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c3">M2</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">w1</font>, <font color="Maroon" title="c7">w2</font> being   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:86_2"/><i><font color="Green" title="E52">A6</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">w1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">w2</font> &amp; <font color="Maroon" title="c6">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M1</font> &amp; <font color="Maroon" title="c7">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c3">M2</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c8">v1</font> being   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E5:86_2"/><i><font color="Green" title="E53">A7</font></i>: 
( <font color="Maroon" title="c6">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c8">v1</font> &amp; <font color="Maroon" title="c8">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M1</font> )
 <b>by </b><i><a class="txt" href="convex4.html#E3:86_2"><i><font color="Green" title="E52">A6</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">v2</font> being   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E7:86_2"/><i><font color="Green" title="E54">A8</font></i>: 
( <font color="Maroon" title="c7">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c9">v2</font> &amp; <font color="Maroon" title="c9">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M2</font> )
 <b>by </b><i><a class="txt" href="convex4.html#E3:86_2"><i><font color="Green" title="E52">A6</font></i></a></i>;<br/>
<a NAME="E8:86_2"/><i><font color="Green" title="E55">A9</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">v2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="convex4.html#E3:86_2"><i><font color="Green" title="E52">A6</font></i></a>, <a class="txt" href="convex4.html#E5:86_2"><i><font color="Green" title="E53">A7</font></i></a>, <a class="txt" href="convex4.html#E7:86_2"><i><font color="Green" title="E54">A8</font></i></a>, <a class="ref" href="clvect_1.html#D2" title="CLVECT_1:def.2">CLVECT_1:def 2</a></i>;<br/>
<a NAME="E9:86_2"/>
<font color="Maroon" title="c8">v1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font>
 
<b>by </b><i><a class="txt" href="convex4.html#E5:86_2"><i><font color="Green" title="E53">A7</font></i></a>, <a class="txt" href="convex4.html#E7:86_2"><i><font color="Green" title="E54">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:86_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font></span>)</span>
 <b>by </b><i><a class="txt" href="convex4.html#E8:86_2"><i><font color="Green" title="E55">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:86"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c3">M2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></i>;<br/>
<b>hence </b><a NAME="E5:86"/>
<font color="Maroon" title="c4">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c3">M2</font></span>)</span>
 <b>by </b><i><a class="txt" href="convex4.html#E2:86"><i><font color="Green" title="E51">A5</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div>
